非线性系统的自适应控制与混沌同步控制
|
摘要
由于实际系统基本都是非线性的,研究有参数不确定性的非线性系统的自适应控制,无论在理论上还是在实际应用中都具有重要的意义.在处理未知定常参数不确定性动态系统的控制问题时,自适应控制成为人们所公认的最有效的方法之一,因为自适应控制能够在线调节控制增益或系统参数,以适应外界环境变化或外界干扰等因素的影响,从而当时间趋于无穷时,实现跟踪误差的渐近收敛.但是对于参数估计来说,只能满足其有界性.而人们总是希望参数估计最好能收敛到其真值,通常在假设持续激励条件下或通过构造系统的强Lyapunov函数(正定、径向无界、导数负定)就可以实现这一目标.近二十年来,混沌控制与同步问题已成为非线性科学领域的热点问题之一,而且在保密通信、信息工程和生命科学等方面显示出巨大的应用前景.自适应控制方法在处理参数不确定性混沌系统的控制与同步问题中也得到了广泛的应用.
基于Lyapunov稳定性理论,本文主要从自适应控制的角度来研究一类严格反馈非线性系统中参数的收敛性和(超)混沌系统的同步问题.主要工作概括如下:
第一,针对一类严格反馈非线性系统,应用Backstepping方法和调节函数法设计系统的控制律和自适应律,在适当的持续激励条件下,构造一个显式、全局的强Lyapunov函数证明了闭环系统的所有信号全局一致有界且跟踪误差和参数估计误差均渐近收敛于零.最后通过数值仿真验证了设计方案的可行性和有效性.
第二,针对两个参数完全未知的不同超混沌系统,基于Lyapunov稳定性理论,设计了相应的控制器和参数自适应律,使得驱动系统和响应系统实现了混合投影同步.最后通过数值仿真验证其有效性.
第三,针对两个带有未知的周期时变参数的不同混沌系统,应用自适应学习控制方法,根据Lyapunov-Krasovskii泛函稳定性理论,构造了微分-差分混合参数学习律和自适应学习控制律,使得两个不同混沌系统的状态误差范数的平方在一个周期区间上的积分意义下实现了渐近同步.该方法成功地应用于Lorenz系统和Chen系统的函数投影同步,最后通过数值仿真验证了该方法的有效性.
The practical systems are almost nonlinear, therefore, the study of adaptive control of nonlinear with uncertain parameters are of great significance both in theory and in practical application. Adaptive control is considered as one of the most effective approaches when solving the control problem of uncertain dynamics with unkonwn constant parameters, because adaptive control can adjust control gains or system parameters online to adapt to the affection factors of environmental changes or disturbance, ect. Hence, asymptotic convergence of tracking error is obtained when time approaches to infinite. However, for estimation of parameters, only boundedness can be guaranteed. It is always hoped that estimation of parameter can converge to its true value, this target can be realized by constructing strong Lyapunov functions (positive definite, radially unbounded and its derivative is negative definite) of systems. During the last two decades, chaos control and synchronization has become one of the hot topics in nonlinear science field, and exhibits wide-scope potential application in various disciplines such as secure communication, information engineering and life science and so on. Adaptive control approach has been widely used to dealing with control and synchronization of chaotic systems with uncertain parameters.
Based on Lyapunov stability theory,this dissertation considers the parameters convergence properties of strict-feedback nonlinear systems and synchronization of (hyper)chaotic systems from an adaptive control viewpoint. The main contributions included in the dissertation are summarized as follows.
Firstly, a strict-feedback nonlinear systems is developed. The design of both the control and the update laws are based on Backstepping techniques and tuning function schemes. Under an appropriate persistency of excitation condition, it is shown that all the closed-loop signals are globally uniformly bounded and both convergence of the tracking error and parameter estimation error to zero asymptotically, by constructing an explicit, global, strong Lyapunov function. The feasibility and effectiveness of the proposed method is illustrated with a simulation example.
Secondly, based on Lyapunov stability theory, control laws and parameter update laws are designed for two different hyperchaotic systems with fully uncertain parameters, the states of the drive system and the response system are ultimately asymptotically hybrid projective synchronization. Simulation results are presented to show the feasibility and effectiveness of the proposed method
At last, a learning control approach is applied to the function projective synchronization of different chaotic systems with unknown periodically time-varying parameters. According to Lyapunov-Krasovskii functional stability theory, a differential-difference mixed-type parametric learning laws and an adaptive learning control laws are constructed to make the states of two different chaotic systems asymptotically synchronized. The approach is successfully applied to the functional projective synchronization between Lorenz system and Chen system. Numerical simulations results are presented to verify the effectiveness of the proposed approach.
基于Lyapunov稳定性理论,本文主要从自适应控制的角度来研究一类严格反馈非线性系统中参数的收敛性和(超)混沌系统的同步问题.主要工作概括如下:
第一,针对一类严格反馈非线性系统,应用Backstepping方法和调节函数法设计系统的控制律和自适应律,在适当的持续激励条件下,构造一个显式、全局的强Lyapunov函数证明了闭环系统的所有信号全局一致有界且跟踪误差和参数估计误差均渐近收敛于零.最后通过数值仿真验证了设计方案的可行性和有效性.
第二,针对两个参数完全未知的不同超混沌系统,基于Lyapunov稳定性理论,设计了相应的控制器和参数自适应律,使得驱动系统和响应系统实现了混合投影同步.最后通过数值仿真验证其有效性.
第三,针对两个带有未知的周期时变参数的不同混沌系统,应用自适应学习控制方法,根据Lyapunov-Krasovskii泛函稳定性理论,构造了微分-差分混合参数学习律和自适应学习控制律,使得两个不同混沌系统的状态误差范数的平方在一个周期区间上的积分意义下实现了渐近同步.该方法成功地应用于Lorenz系统和Chen系统的函数投影同步,最后通过数值仿真验证了该方法的有效性.
The practical systems are almost nonlinear, therefore, the study of adaptive control of nonlinear with uncertain parameters are of great significance both in theory and in practical application. Adaptive control is considered as one of the most effective approaches when solving the control problem of uncertain dynamics with unkonwn constant parameters, because adaptive control can adjust control gains or system parameters online to adapt to the affection factors of environmental changes or disturbance, ect. Hence, asymptotic convergence of tracking error is obtained when time approaches to infinite. However, for estimation of parameters, only boundedness can be guaranteed. It is always hoped that estimation of parameter can converge to its true value, this target can be realized by constructing strong Lyapunov functions (positive definite, radially unbounded and its derivative is negative definite) of systems. During the last two decades, chaos control and synchronization has become one of the hot topics in nonlinear science field, and exhibits wide-scope potential application in various disciplines such as secure communication, information engineering and life science and so on. Adaptive control approach has been widely used to dealing with control and synchronization of chaotic systems with uncertain parameters.
Based on Lyapunov stability theory,this dissertation considers the parameters convergence properties of strict-feedback nonlinear systems and synchronization of (hyper)chaotic systems from an adaptive control viewpoint. The main contributions included in the dissertation are summarized as follows.
Firstly, a strict-feedback nonlinear systems is developed. The design of both the control and the update laws are based on Backstepping techniques and tuning function schemes. Under an appropriate persistency of excitation condition, it is shown that all the closed-loop signals are globally uniformly bounded and both convergence of the tracking error and parameter estimation error to zero asymptotically, by constructing an explicit, global, strong Lyapunov function. The feasibility and effectiveness of the proposed method is illustrated with a simulation example.
Secondly, based on Lyapunov stability theory, control laws and parameter update laws are designed for two different hyperchaotic systems with fully uncertain parameters, the states of the drive system and the response system are ultimately asymptotically hybrid projective synchronization. Simulation results are presented to show the feasibility and effectiveness of the proposed method
At last, a learning control approach is applied to the function projective synchronization of different chaotic systems with unknown periodically time-varying parameters. According to Lyapunov-Krasovskii functional stability theory, a differential-difference mixed-type parametric learning laws and an adaptive learning control laws are constructed to make the states of two different chaotic systems asymptotically synchronized. The approach is successfully applied to the functional projective synchronization between Lorenz system and Chen system. Numerical simulations results are presented to verify the effectiveness of the proposed approach.
引文
[1]李言俊,张科.自适应控制理论及应用[M].西安:西北工业大学出版社,2005.
[2] Kokotovic K P V. Foundations of Adaptive Control[M]. New York: Springer- Verlag, 1991.
[3] Kanellakopoulos I., Kokotovic P.V. and Morse A.S. Systematic design of adaptive controllers for feedback linearizable systems[J]. IEEE Transaction on Automatic Control, 1991, 36(11): 1241-1253.
[4] Krstic M., Kanellkopoulos I.and Kokotovic P.V. Adaptive nonlinear control without over-parameterization[J]. Systems and Control Letters, 1992, 19(3): 177-185.
[5] Kanellakopoulos I., Kokotovic P.V. and Middleton R.H. Observer-based adaptive output-feedback control of nonlinear systems under matching condition [C]. Proceeding of American Control Conference, 1990, 549-555.
[6] Kanellakopoulos I., Kokotovic P.V. and Middleton R.H. Indirect adaptive output-feedback control of a class of nonlinear systems[C]. Proceeding of 29th Conference on Decision and Control. 1990: 2714-2719.
[7] Kanellakopoulos I., Kokotovic P.V. and Morse A.S. Adaptive output-feedback control of systems with output nonlinearities[J] . IEEE Transaction on Automatic Control, 1992, 37(11): 1666-1682.
[8] Marrino R. and Tomei P. Global adaptive output-feedback control of nonlinear systems, part I: linear parameterization[J]. IEEE Transaction on Automatic Control, 1993, 38(1): 17-32.
[9] Marrino R.and Tomei P. Global adaptive output-feedback control of nonlinear systems, part II: linear parameterization[J]. IEEE Transaction on Automatic Control, 1993, 38(1): 33-48.
[10]陈卫田,施颂椒,张钟俊.不确定非线性系统的鲁棒自适应控制[J].上海交通大学学报,1998, 32(6): 88-93.
[11]黄长水,阮荣耀.一类不确定性非线性系统的鲁棒自适应控制[J].自动化学报,2001, 27(1): 82-88.
[12]Han H G, Su C Y, Stepanenko Y. Adaptive Control of a Class of Nonlinear Systems with Nonlinear Parameterized Fuzzy Approximators[J]. IEEETransactions on Fuzzy Systems, 2001, 9(2): 315-323.
[13]Boskovic J. D. Adaptive control of a class of nonlinear parameterized plant[J]. IEEE Transactions on Automatic Control, 1998, 43(7): 930-934.
[14]Ye X D. Global adaptive control of nonlinear parameterized systems[J]. IEEE Transactions on Automatic Control, 2003, 48(1): 169-173.
[15]Qu Z H, Hull R. A. and Wang J. Global stabilizing adaptive control design for nonlinearly parameterized systems[J]. IEEE Transactions on Automatic Control, 2006, 51(6): 1073-1079.
[16]Mazenc F. Strict Lyapunov functions for time-varying systems[J]. Automatica, 2003, 39(2): 349-353.
[17]Mazenc F., Nesic D. Strong Lyapunov functions for systems satisfying the conditions of LaSalle[J]. IEEE Transactions on Automatic Control, 2004, 49(6): 1026-1030.
[18]Mazenc F., Malisoff M. Further constructions of control-Lyapunov functions and stabilizing feedbacks for systems satisfying the Jurdjevic-Quinn conditions[J]. IEEE Transactions on Automatic Control, 2006, 51(2): 360-365.
[19]Mazenc F., Malisoff M. Lyapunov functions constructions for slowly time-varying systems[J]. Mathematics of Control Signals and Systems, 2007, 19(1): 1-21.
[20]Mazenc F., Nesic D. Lyapunov functions for time varying systems satisfying generalized contions of Matrosov theorem[J]. Mathematics of Control Signals and Systems, 2007, 19(2): 151-182.
[21]Mazenc F., Malisoff M., Bernard O. A simplified design for strict Lyapunov functions under Matrosov conditions[J]. IEEE Transactions on Automatic Control, 2009, 54(1): 177-183.
[22]Mazenc F., Queiroz M, Malisoff M.. Uniform global asymptotic stability of a class of adaptively controlled nonlinear systems[J]. IEEE Transactions on Automatic Control, 2009, 54(5): 1152-1158.
[23]Fauboug L., Pomet J. B. Control Lyapunov functions for homogeneou“Jurdjevic-Quinn”systems[J]. European Series in Applied and Industrial Mathematics: Control Optimisation and Calculus of Variations, 2000, 5: 293-311.
[24]Mazenc F., Malisoff M. Strict Lyapunov functions constructions under LaSalle conditions with an application to Lotka-Volterra systems[J]. IEEE Transactions on Automatic Control, 2010, 55(4): 841-854.
[25]Michael G. Rosenblum, Arkady S. Pikovsky and Jürgen Kurths. Phase synchronization of chaotic oscillators[J]. Physical Review Letters, 1996, 76(11): 1804-1807.
[26]Shahverdiev E. M., Sivaprakasam S. and Shore K.A. Lag synchronization in time-delayed systems[J]. Physics Letters A, 2002, 292(6): 320-324.
[27]Henry D. I. Abarbanel, Nikolai F. Rulkov and Mikhail M. Sushchik. Generalized synchronization of chaos: The auxiliary system approach[J]. Physical Review E, 1996, 53(5): 4528-4535.
[28]Pecora L. M., Carroll T. L. Synchronization in chaotic systems[J]. Physical Review Letters, 1990, 64(8): 821–824.
[29]Pecora L. M., Carroll T. L. Carroll. Driving systems with chaotic signals[J]. Physical Review A, 1991, 44(4): 2374-2383.
[30]Kocarev L. and Parlitz U. General approach for chaotic synchronization with applications to communication[J]. Physical Review Letters, 1995, 74(25): 5028-5031.
[31]Guo-Ping Jiang and Wallace K. S. Tang. A global synchronization criterion for coupled chaotic systems via unidirectional linear error feedback approach[J]. International Journal of Bifurcation and Chaos, 2002, 12(10): 2239-2253.
[32]高金峰,罗先觉,马西奎.控制与同步连续时间混沌系统的非线性反馈方法[J].物理学报, 1999, 48(9): 1618-1627.
[33]Huberman B, ALumer E.,Dynamics of adaptive systems[J]. IEEE Transaction Circuits and Systems I, 1990, 37(4): 547-550.
[34]Sinha S., Ramaswany R. and Rao J. Adaptive Control in nonlinear dynamics[J]. Physica D, 1990, 43(1): 115-128.
[35]John J. K. and Amritkar R. E. Synchronization of unstable orbits using adaptive control[J]. Physical Review E, 1994, 49(6): 4843-4848.
[36]Yu Y G, Zhang S C. Controlling uncertain Lüsystem using backstepping design[J]. Chaos, Solitons and Fractals, 2003, 15(5): 897-902.
[37]Luo R Z. Impulsive control and synchronization of a new chaotic system[J]. Physics Letters A, 2008, 372: 648–653.
[38]Jang M J, Chen C L, Chen C K, Sliding mode control of hyperchaos in R?sler systems[J]. Chaos, Solitons and Fractals, 2002, 14: 1465–1476.
[39]Park J H. Chaos synchronization of between two different chaotic dynamical systems [J]. Chaos, Solitons and Fractals, 2006, 27: 549–554.
[40]Lei Y M, Xu W, Zheng H C. Synchronization of two chaotic nonlinear gyrosusing active control[J]. Physics Letters A, 2005, 343: 153–158.
[41]Mainieri R., Rehacek J. Projective synchronization in the three-dimensional chaotic systems[J]. Physical Review Letters, 1999, 82: 3042–3045.
[42]Yan Janping and Li Changpin. Generalized projective synchronization of a unified chaotic system[J]. Chaos, Solitons and Fractals, 2005, 26(4): 1119-1124.
[43]Hu M F, Xu Z Y, Zhang R, etc. Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems[J]. Physics Letters A, 2007, 361(3): 231–237.
[44]Ioannou P A, Sun J. Robust Adaptive Control. Upper Saddle River[M], NJ: Prentice-Hall, 1996.
[45]Krstic M. Invariant manifolds and asymptotic properties of adaptive nonlinear stabilizers[J]. IEEE Transactions on Automatic Control, 1996, 41(6): 817-829.
[46]Narendra K S, Annaswamy A M. Persistent excitation in adaptive systems[J]. International Journal of Control, 1987, 45(1): 127-160.
[47]Lin J S, Kanellakopoulos I. Nonlinearities enhance parameter convergence in output-feedback systems[J]. IEEE Transactions on Automatic Control, 1998, 43(2): 204-222.
[48]Lin J S, Kanellakopoulos I. Nonlinearities enhance parameter convergence in strict feedback systems[J]. IEEE Transactions on Automatic Control, 1999, 44(1): 89-94.
[49]Adetola V, Guay M. Finite-time parameter estimation in adaptive control of nonlinear systems[J]. IEEE Transactions on Automatic Control, 2008, 53(3): 807-811.
[50]Adetola V, Guay M. Performance improvement in adaptive control of nonlinear system[C]. Proceeding of American Control Conference, Hyatt Regency Riverfront, St. Louis, MO, USA, June 10-12, 2009: 5610-5615.
[51]Loría A, Kelly R, Teel A R. Uniform parametric convergence in the adaptive control of mechanical systems[J]. European Journal of Control, 2005, 11: 87-100.
[52]Loría A, Panteley E, Popovic D, etc.δ-persistency of excitation: A necessary and sufficient condition for uniform attractivity[C]. Proceedings of the 41st IEEE Conference on Decision and Control. Las Vegas, Nevada USA, 2002: 3506-3511.
[53]Krstic M, Kanellakopoulos I and Kokotovic P. Nonlinear and Adaptive Control Design[M]. New York: Wiley, 1995.
[54]R?sler O.E. An equation for hyperchaos[J]. Physics Letters A, 1979, 71: 155–157.
[55]Li Y, Tang W.K.S., Chen G. Generating hyperchaos via state feedback control[J]. International Journal of Bifurcation and Chaos, 2005, 15: 3367–3375.
[56]Li Y, Tang W.K.S., Chen G.. Chen. Hyperchaos evolved from the generalized Lorenz equation[J]. International Journal Circuit Theory Applications, 2005, 33: 235–251.
[57]Chen A, Lu J, LüJ, etc. Generating hyperchaotic Lüattractor via state feedback control[J]. Physica A: Statistical Mechanics and its Applications, 2006, 364: 103–110.
[58]Vincent U. E. Synchronization of identical and non-identical 4-D chaotic systems using active control[J]. Chaos, Solitons and Fractals, 2008, 37: 1065-1075.
[59]Chee C Y, Xu D L. Chaos-based M-nary digital communication technique using controlled projective synchronization[C]. IEE Proceeding Circuits Devices System. 2006, 153(4): 357-360.
[60]Wen G L, Xu D L. Observed-based control for full state projective synchronization of a general class of chaotic maps in any dimension[J]. Physics Letters A, 2004, 333: 420-425.
[61]Hung Y C, Hwang C C, Liao T L, etc. Generalized projective synchronization of chaotic systems with unknown dead-zone input: Observed- basedapproach[J]. Chaos , 2006, 16(3): 3125-3133.
[62]Hu M F, Xu Z Y, Yang Y Q. Projective cluster synchronization in drive- response dynamical networks[J]. Physica A, 2008, 387: 3759-3768.
[63]Li G H. Modified projective synchronization of chaotic system[J]. Chaos, Solitons and Fractals, 2007, 32(5): 1786-1790.
[64]Tang Y, Fang J A. General method for modified projective synchronization of hyperchaotic systems with known or unknown parameters[J]. Physics Letters A, 2008, 372: 1816-1826.
[65]Hu M F, Xu Z Y, Zhang R, etc. Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order[J]. Physics Letters A, 2007, 365: 315-327.
[66]Hu M F, Xu Z Y, Zhang R. Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems[J]. Communications in Nonlinear Science and Numerical Simulation, 2008, 13: 456-464.
[67]Zheng Song , Dong Gaogao , Bi Qinsheng. A new hyperchaotic system and its synchronization[J]. Applied Mathematics and Computation, 2010, 215: 3192-3200.
[68]Zhu Congxu. Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters[J]. Applied Mathematics and Computation, 2009, 215: 557–561.
[69]Chen Z, Yang Y, Qi G, etc. A novel hyperchaos system only with one equilibrium[J]. Physics Letters A, 2007, 360: 696–701.
[70]Wang X, Wang M. A hyperchaos generated from Lorenz system[J]. Physica A: Statistical Mechanics and its Applications, 2008, 387: 3751–3758.
[71]Du H Y, Zeng Q S and Wang C G.. Function projective synchronization of different chaotic systems with uncertain parameters[J]. Physics Letters A, 2008, 372: 5402-5410.
[72]Salarieh H,Shahrokhi M. Adaptive synchronization of two different chaotic systems with time varying unknown parameters[J]. Chaos, Solitons and Fractals, 2008, 37(1): 125-136.
[73]Sun F Y, Zhao Y and Zhou T S. Identify fully uncertain parameters and design controllers based on synchronization[J]. Chaos, Solitons and Fractals, 2007, 34(5): 1677-1682.
[74]Park J H. Adaptive modified projective synchronization of a unified chaotic system with all uncertain parameter[J]. Chaos, Solitons and Fractals, 2007, 34(5): 1552-1559.
[75]Xu J X, Yan R. Synchronization of chaotic systems via learning control[J]. International Journal of Bifurcation and Chaos, 2005, 15(2): 4035-4041.
[76]Song Y X,Yu X H,G R Chen,etc. Time delayed repetitive learning control for chaotic systems[J]. International Journal of Bifurcation and Chaos, 2002, 12(5): 1057-1065.
[77]Sun Yun-Ping, Li Jun-Min, Wang Jiang-An and Wang Hui-Lin. Generalized projective synchronization of chaotic systems via adaptive learning control[J]. Chinese Physical B, 2010, 19(2): 020505(1-8).
[2] Kokotovic K P V. Foundations of Adaptive Control[M]. New York: Springer- Verlag, 1991.
[3] Kanellakopoulos I., Kokotovic P.V. and Morse A.S. Systematic design of adaptive controllers for feedback linearizable systems[J]. IEEE Transaction on Automatic Control, 1991, 36(11): 1241-1253.
[4] Krstic M., Kanellkopoulos I.and Kokotovic P.V. Adaptive nonlinear control without over-parameterization[J]. Systems and Control Letters, 1992, 19(3): 177-185.
[5] Kanellakopoulos I., Kokotovic P.V. and Middleton R.H. Observer-based adaptive output-feedback control of nonlinear systems under matching condition [C]. Proceeding of American Control Conference, 1990, 549-555.
[6] Kanellakopoulos I., Kokotovic P.V. and Middleton R.H. Indirect adaptive output-feedback control of a class of nonlinear systems[C]. Proceeding of 29th Conference on Decision and Control. 1990: 2714-2719.
[7] Kanellakopoulos I., Kokotovic P.V. and Morse A.S. Adaptive output-feedback control of systems with output nonlinearities[J] . IEEE Transaction on Automatic Control, 1992, 37(11): 1666-1682.
[8] Marrino R. and Tomei P. Global adaptive output-feedback control of nonlinear systems, part I: linear parameterization[J]. IEEE Transaction on Automatic Control, 1993, 38(1): 17-32.
[9] Marrino R.and Tomei P. Global adaptive output-feedback control of nonlinear systems, part II: linear parameterization[J]. IEEE Transaction on Automatic Control, 1993, 38(1): 33-48.
[10]陈卫田,施颂椒,张钟俊.不确定非线性系统的鲁棒自适应控制[J].上海交通大学学报,1998, 32(6): 88-93.
[11]黄长水,阮荣耀.一类不确定性非线性系统的鲁棒自适应控制[J].自动化学报,2001, 27(1): 82-88.
[12]Han H G, Su C Y, Stepanenko Y. Adaptive Control of a Class of Nonlinear Systems with Nonlinear Parameterized Fuzzy Approximators[J]. IEEETransactions on Fuzzy Systems, 2001, 9(2): 315-323.
[13]Boskovic J. D. Adaptive control of a class of nonlinear parameterized plant[J]. IEEE Transactions on Automatic Control, 1998, 43(7): 930-934.
[14]Ye X D. Global adaptive control of nonlinear parameterized systems[J]. IEEE Transactions on Automatic Control, 2003, 48(1): 169-173.
[15]Qu Z H, Hull R. A. and Wang J. Global stabilizing adaptive control design for nonlinearly parameterized systems[J]. IEEE Transactions on Automatic Control, 2006, 51(6): 1073-1079.
[16]Mazenc F. Strict Lyapunov functions for time-varying systems[J]. Automatica, 2003, 39(2): 349-353.
[17]Mazenc F., Nesic D. Strong Lyapunov functions for systems satisfying the conditions of LaSalle[J]. IEEE Transactions on Automatic Control, 2004, 49(6): 1026-1030.
[18]Mazenc F., Malisoff M. Further constructions of control-Lyapunov functions and stabilizing feedbacks for systems satisfying the Jurdjevic-Quinn conditions[J]. IEEE Transactions on Automatic Control, 2006, 51(2): 360-365.
[19]Mazenc F., Malisoff M. Lyapunov functions constructions for slowly time-varying systems[J]. Mathematics of Control Signals and Systems, 2007, 19(1): 1-21.
[20]Mazenc F., Nesic D. Lyapunov functions for time varying systems satisfying generalized contions of Matrosov theorem[J]. Mathematics of Control Signals and Systems, 2007, 19(2): 151-182.
[21]Mazenc F., Malisoff M., Bernard O. A simplified design for strict Lyapunov functions under Matrosov conditions[J]. IEEE Transactions on Automatic Control, 2009, 54(1): 177-183.
[22]Mazenc F., Queiroz M, Malisoff M.. Uniform global asymptotic stability of a class of adaptively controlled nonlinear systems[J]. IEEE Transactions on Automatic Control, 2009, 54(5): 1152-1158.
[23]Fauboug L., Pomet J. B. Control Lyapunov functions for homogeneou“Jurdjevic-Quinn”systems[J]. European Series in Applied and Industrial Mathematics: Control Optimisation and Calculus of Variations, 2000, 5: 293-311.
[24]Mazenc F., Malisoff M. Strict Lyapunov functions constructions under LaSalle conditions with an application to Lotka-Volterra systems[J]. IEEE Transactions on Automatic Control, 2010, 55(4): 841-854.
[25]Michael G. Rosenblum, Arkady S. Pikovsky and Jürgen Kurths. Phase synchronization of chaotic oscillators[J]. Physical Review Letters, 1996, 76(11): 1804-1807.
[26]Shahverdiev E. M., Sivaprakasam S. and Shore K.A. Lag synchronization in time-delayed systems[J]. Physics Letters A, 2002, 292(6): 320-324.
[27]Henry D. I. Abarbanel, Nikolai F. Rulkov and Mikhail M. Sushchik. Generalized synchronization of chaos: The auxiliary system approach[J]. Physical Review E, 1996, 53(5): 4528-4535.
[28]Pecora L. M., Carroll T. L. Synchronization in chaotic systems[J]. Physical Review Letters, 1990, 64(8): 821–824.
[29]Pecora L. M., Carroll T. L. Carroll. Driving systems with chaotic signals[J]. Physical Review A, 1991, 44(4): 2374-2383.
[30]Kocarev L. and Parlitz U. General approach for chaotic synchronization with applications to communication[J]. Physical Review Letters, 1995, 74(25): 5028-5031.
[31]Guo-Ping Jiang and Wallace K. S. Tang. A global synchronization criterion for coupled chaotic systems via unidirectional linear error feedback approach[J]. International Journal of Bifurcation and Chaos, 2002, 12(10): 2239-2253.
[32]高金峰,罗先觉,马西奎.控制与同步连续时间混沌系统的非线性反馈方法[J].物理学报, 1999, 48(9): 1618-1627.
[33]Huberman B, ALumer E.,Dynamics of adaptive systems[J]. IEEE Transaction Circuits and Systems I, 1990, 37(4): 547-550.
[34]Sinha S., Ramaswany R. and Rao J. Adaptive Control in nonlinear dynamics[J]. Physica D, 1990, 43(1): 115-128.
[35]John J. K. and Amritkar R. E. Synchronization of unstable orbits using adaptive control[J]. Physical Review E, 1994, 49(6): 4843-4848.
[36]Yu Y G, Zhang S C. Controlling uncertain Lüsystem using backstepping design[J]. Chaos, Solitons and Fractals, 2003, 15(5): 897-902.
[37]Luo R Z. Impulsive control and synchronization of a new chaotic system[J]. Physics Letters A, 2008, 372: 648–653.
[38]Jang M J, Chen C L, Chen C K, Sliding mode control of hyperchaos in R?sler systems[J]. Chaos, Solitons and Fractals, 2002, 14: 1465–1476.
[39]Park J H. Chaos synchronization of between two different chaotic dynamical systems [J]. Chaos, Solitons and Fractals, 2006, 27: 549–554.
[40]Lei Y M, Xu W, Zheng H C. Synchronization of two chaotic nonlinear gyrosusing active control[J]. Physics Letters A, 2005, 343: 153–158.
[41]Mainieri R., Rehacek J. Projective synchronization in the three-dimensional chaotic systems[J]. Physical Review Letters, 1999, 82: 3042–3045.
[42]Yan Janping and Li Changpin. Generalized projective synchronization of a unified chaotic system[J]. Chaos, Solitons and Fractals, 2005, 26(4): 1119-1124.
[43]Hu M F, Xu Z Y, Zhang R, etc. Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems[J]. Physics Letters A, 2007, 361(3): 231–237.
[44]Ioannou P A, Sun J. Robust Adaptive Control. Upper Saddle River[M], NJ: Prentice-Hall, 1996.
[45]Krstic M. Invariant manifolds and asymptotic properties of adaptive nonlinear stabilizers[J]. IEEE Transactions on Automatic Control, 1996, 41(6): 817-829.
[46]Narendra K S, Annaswamy A M. Persistent excitation in adaptive systems[J]. International Journal of Control, 1987, 45(1): 127-160.
[47]Lin J S, Kanellakopoulos I. Nonlinearities enhance parameter convergence in output-feedback systems[J]. IEEE Transactions on Automatic Control, 1998, 43(2): 204-222.
[48]Lin J S, Kanellakopoulos I. Nonlinearities enhance parameter convergence in strict feedback systems[J]. IEEE Transactions on Automatic Control, 1999, 44(1): 89-94.
[49]Adetola V, Guay M. Finite-time parameter estimation in adaptive control of nonlinear systems[J]. IEEE Transactions on Automatic Control, 2008, 53(3): 807-811.
[50]Adetola V, Guay M. Performance improvement in adaptive control of nonlinear system[C]. Proceeding of American Control Conference, Hyatt Regency Riverfront, St. Louis, MO, USA, June 10-12, 2009: 5610-5615.
[51]Loría A, Kelly R, Teel A R. Uniform parametric convergence in the adaptive control of mechanical systems[J]. European Journal of Control, 2005, 11: 87-100.
[52]Loría A, Panteley E, Popovic D, etc.δ-persistency of excitation: A necessary and sufficient condition for uniform attractivity[C]. Proceedings of the 41st IEEE Conference on Decision and Control. Las Vegas, Nevada USA, 2002: 3506-3511.
[53]Krstic M, Kanellakopoulos I and Kokotovic P. Nonlinear and Adaptive Control Design[M]. New York: Wiley, 1995.
[54]R?sler O.E. An equation for hyperchaos[J]. Physics Letters A, 1979, 71: 155–157.
[55]Li Y, Tang W.K.S., Chen G. Generating hyperchaos via state feedback control[J]. International Journal of Bifurcation and Chaos, 2005, 15: 3367–3375.
[56]Li Y, Tang W.K.S., Chen G.. Chen. Hyperchaos evolved from the generalized Lorenz equation[J]. International Journal Circuit Theory Applications, 2005, 33: 235–251.
[57]Chen A, Lu J, LüJ, etc. Generating hyperchaotic Lüattractor via state feedback control[J]. Physica A: Statistical Mechanics and its Applications, 2006, 364: 103–110.
[58]Vincent U. E. Synchronization of identical and non-identical 4-D chaotic systems using active control[J]. Chaos, Solitons and Fractals, 2008, 37: 1065-1075.
[59]Chee C Y, Xu D L. Chaos-based M-nary digital communication technique using controlled projective synchronization[C]. IEE Proceeding Circuits Devices System. 2006, 153(4): 357-360.
[60]Wen G L, Xu D L. Observed-based control for full state projective synchronization of a general class of chaotic maps in any dimension[J]. Physics Letters A, 2004, 333: 420-425.
[61]Hung Y C, Hwang C C, Liao T L, etc. Generalized projective synchronization of chaotic systems with unknown dead-zone input: Observed- basedapproach[J]. Chaos , 2006, 16(3): 3125-3133.
[62]Hu M F, Xu Z Y, Yang Y Q. Projective cluster synchronization in drive- response dynamical networks[J]. Physica A, 2008, 387: 3759-3768.
[63]Li G H. Modified projective synchronization of chaotic system[J]. Chaos, Solitons and Fractals, 2007, 32(5): 1786-1790.
[64]Tang Y, Fang J A. General method for modified projective synchronization of hyperchaotic systems with known or unknown parameters[J]. Physics Letters A, 2008, 372: 1816-1826.
[65]Hu M F, Xu Z Y, Zhang R, etc. Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order[J]. Physics Letters A, 2007, 365: 315-327.
[66]Hu M F, Xu Z Y, Zhang R. Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems[J]. Communications in Nonlinear Science and Numerical Simulation, 2008, 13: 456-464.
[67]Zheng Song , Dong Gaogao , Bi Qinsheng. A new hyperchaotic system and its synchronization[J]. Applied Mathematics and Computation, 2010, 215: 3192-3200.
[68]Zhu Congxu. Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters[J]. Applied Mathematics and Computation, 2009, 215: 557–561.
[69]Chen Z, Yang Y, Qi G, etc. A novel hyperchaos system only with one equilibrium[J]. Physics Letters A, 2007, 360: 696–701.
[70]Wang X, Wang M. A hyperchaos generated from Lorenz system[J]. Physica A: Statistical Mechanics and its Applications, 2008, 387: 3751–3758.
[71]Du H Y, Zeng Q S and Wang C G.. Function projective synchronization of different chaotic systems with uncertain parameters[J]. Physics Letters A, 2008, 372: 5402-5410.
[72]Salarieh H,Shahrokhi M. Adaptive synchronization of two different chaotic systems with time varying unknown parameters[J]. Chaos, Solitons and Fractals, 2008, 37(1): 125-136.
[73]Sun F Y, Zhao Y and Zhou T S. Identify fully uncertain parameters and design controllers based on synchronization[J]. Chaos, Solitons and Fractals, 2007, 34(5): 1677-1682.
[74]Park J H. Adaptive modified projective synchronization of a unified chaotic system with all uncertain parameter[J]. Chaos, Solitons and Fractals, 2007, 34(5): 1552-1559.
[75]Xu J X, Yan R. Synchronization of chaotic systems via learning control[J]. International Journal of Bifurcation and Chaos, 2005, 15(2): 4035-4041.
[76]Song Y X,Yu X H,G R Chen,etc. Time delayed repetitive learning control for chaotic systems[J]. International Journal of Bifurcation and Chaos, 2002, 12(5): 1057-1065.
[77]Sun Yun-Ping, Li Jun-Min, Wang Jiang-An and Wang Hui-Lin. Generalized projective synchronization of chaotic systems via adaptive learning control[J]. Chinese Physical B, 2010, 19(2): 020505(1-8).